Three essays on collusion and mergers

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EUROPEAN UNIVERSITY INSTITUTE

Department of Economics

T hree E ssays on C ollusion and M ergers

Helder Vasconcelos

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Thesis submitted for assessment with a view to obtaining

the degree o f Doctor o f the European University Institute

Florence

October 2002

European University Institute 3 0001 0042 4159 4 y 9 ■ v 1 o v o EU RO PEAN UNIVERSITY IN ST IT U T E D ep artm en t o f Economics

Three Essays on Collusion and Mergers

Helder Vasconcelos

L I B

The Thesis Committee consists of:

P ro f Pierpaolo Battigalli, Università Bocconi

Luis Cabral, Leonard Stem School o f Business, NYU Massimo Motta, EUI, Supervisor

Karl Schlag, EUI

John Sutton, London School o f Economics

To Anabela

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Acknowledgments

In the process of writing this dissertation, I have accumulated many debts. Time has come to express my gratitude to those people and institutions that have contributed, in a variety of ways, for the preparation of this work.

M y first debt is to my supervisor, Massimo Motta. I could not have been more fortunate with my supervisor. M assimo’s supervision was outstanding: very knowledgeable and demanding. His permanent availability to discuss my research work and very fast feedback on preliminary drafts o f my research papers should also be highlighted. Working under his supervision proved to be extremely enriching and stimulating.

My second debt is to Pierpaolo Battigalli. His co-supervision was also ex­ cellent. He very often read my work with patience and provided me very detailed comments on the more technical parts of the chapters, which substantially improved them. With him I learned how important it is to use clear and precise notation in economic modelling.

My special thanks are also due to Karl Schlag. Even though he was not for­ mally my thesis advisor, I benefited enormously from several discussions in his office about my thesis two last chapters. I am grateful for a substantial amount of work and very useful suggestions on those chapters.

Many other people have helped to improve the quality of this thesis, by com ­ menting on specific chapters of this dissertation. I am particularly indebted to Pedro

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Pita Barros, Francis Bloch, Natalia Fabra, Chiara Fumagalli, Joseph E. Harrington, Jr., Andrea Ichino, Soren Johansen, Patrick Rey, Margaret Slade and Larry Samuel- son. Thanks are also due to Robert Porter for providing me with the data set on the Joint Executive Committee, which I use in the empirical part of chapter two.

On the institutional side, a very warm acknowledgment goes to the Economics o f Industry Group (STICERD) at the London School of Economics and Political Science, for the hospitality provided while I spent four months as a visiting re­ search student. There I started the last chapter o f this thesis under Professor John Sutton’s supervision. I am indebted for his encouragement and invaluable sug­ gestions. Moreover, I would like to thank Professor Sutton for having welcomed me as one o f

his

students from the very first minute. Financial support from the Portuguese Ministry of Foreign Affairs, the Portuguese Ministry o f Science and Technology and the European University Institute is gratefully acknowledged.

Sem inar participants at the European University Institute, Research Institute for Industrial Economics (IUI, Stockholm), Universität Pompeu Fabra, the Univer­ sity o f Warwick, University of Lausanne, ISEG (Lisbon), Universidade do Minho and Universidade Nova de Lisboa as well as conference participants at the 1999 C.I.E. Sum m er School (University o f Copenhagen), Second CEPR Conference in A pplied Industrial Organization, the 2000 and 2001 annual conferences o f the

Eu­

ropean Economic Association

(EEA), 27th Conference o f the

European Association

- u u u u u u u :

Acknow] edgments v

for Research in Industrial Economics

(EARIE) and ASSET 2001 Euroconference have commented on various stages of the work presented here.

I am also grateful to Jacqueline Bourgonje, Jessica Spataro and Marcia Gastaldo for their very friendly support and invaluable administrative help over the last four years.

Finally, a very special mention to those who have provided me the most sup­ port: my parents, my beloved sister Sandra and my wife Anabela. My parents and sister encouraged me to follow my own decision to perform this project even if this meant having to live apart from each other for four years time. I have no words to express my gratitude to them. They were simply exceptional, as always. M y great­ est debt is, however, to my wife Anabela. She sacrificed her professional career to be in Florence with me and provided me with her love and continuous support throughout the whole project. This thesis is dedicated to her.

I L L J

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Contents

Introductory overview... 1

1 Entry Effects on Cartel Stability and the Joint Executive Committee 7

1.1 Introduction... 7

1.2 The basic m odel...10

1.3 Considering e n try ... 16

1.3.1 The new model structure...16

1.3.2 Case 1. Cartel Breakdown ... 19

1.3.3 Case 2. Accommodation... 25

1.3.4 Stability discussion... 30

1.4 The Joint Executive C o m m ittee... 33

1.5 The d a t a ... 34

1.6 The econometric m odel...37

1.7 Conclusion... 42

2 Tacit Collusion, Cost Asymmetries and M erg ers... 44

2.1 Introduction... 44

2.2 The m o d e l... 48

2.3 The analysis o f perfect efficient collusion...52

2.3.1 Collusive profits... 53

2.3.2 Deviation profits... 54

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Contents _vii

2.3.3 Distribution of assets and scope for collusion ...55

2.4 Perfect non-efficient collusion...76

2.5 C onclusion...80

2. A The Cournot equilibrium...83

3 Towards a Characterization of the Upper Bound to

Concentration in Endogenous Sunk Cost Industries... 84

3.1 Introduction... 84

3.2 The basic m odel...88

3.2.1 The g a m e ... 89

3.2.2 Equilibrium analysis... 92

3.3 C onclusion... 129

3. A Stability requirem ents...131

3.B Upper bound to the number entrants anticipating a duopoly coalition structure...134

Bibliography... 135

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Introductory overview

This thesis aims at contributing to the analysis o f the issues o f collusion and mergers from an industrial organization perspective. The thesis is composed of three main chapters. In what follows each of those chapters will be discussed in greater detail. In particular, I will briefly describe the model considered in each chapter, its relation to the literature, and the contribution each chapter makes to the existing literature.

Chapter one deals with the issue o f entry and collusion both theoretically and em ­ pirically. In particular, it re-considers Green and Porter’s (1984) model o f collusion with imperfect monitoring (in the price version, as in Tirole’s (1988) textbook). In this model with homogeneous goods, a firm can either charge the monopoly price to share the mar­ ket with other firms or secretly undercut its rivals to get the whole market. Since market demand is affected by stochastic factors, firm’s demand could be zero even when no firm is deviating. Given that the only available information to each firm is its own sales, it is difficult for them to infer whether a low demand is caused by a secret price cut or it is just due to a “bad” demand shock. In equilibrium, the firms need to start a price-war (i.e., charge the competitive price) when the demand is zero to maintain the incentive to charge the monopoly price in the collusive phases. As a result, a price-war occurs with positive probability as an equilibrium phenomenon.

New in this chapter is the existence of potential entrants in the cartel. The chapter examines the structure and success of a cartel, of the type described above, when there ex- ists a pool o f potential entrants. To do so, a no-entry condition has to be taken into account

1

Introductory overview

2

in addition to the standard incentive compatibility constraint to maintain the collusive price level in the collusive phases. A potential entrant is assumed to enter the market (and the cartel) in any period (which is an irreversible decision) if its benefit from a continuation equilibrium exceeds the initial entry fixed sunk cost. Hence, how incumbent firms react to an entrant is important to determine how severe the no-entry constraint is. Two differ­ ent incum bents’ response to entry are considered: (t) reversion to finite-punishment phase, and (u ) accommodation of entrants. Naturally, it is shown that the no-entry constraint is most difficult to be satisfied when incumbent firms accommodate the entrant, which was actually the case for the entry episodes fo r the nineteenth-century US railroad cartel whose pricing decisions are discussed in the applied part o f the chapter.

The chapter shows that the optimal length o f a price w ar increases as the number of firms in the cartel agreement increases if and only if the entry cost is so high that the no­ entry constraint is not binding. The intuition that underlies this result is simple. When there are more firms in the agreement, each firm has a smaller share o f the market. Then, first, the gain from secret price cutting becom es larger as a deviating firm can obtain a larger share of the market, and, second, each firm ’s continuation payoff in the collusive equilibrium decreases, which reduces the impact o f any finite-period punishment. Combining these two effects, one concludes that the duration of a “price-war” has to

increase

to maintain the incentive for the firms to stick to th e collusive price.

Since the demand shock is assum ed to be independent and identically distributed (i.i.d.), the probability to trigger a price-w ar is the same for every period in a collusive phase. Given this, it is also shown that the percentage of periods spent in a price-war

in-. Introductory overview 3

creases with the length of the punishment phase, and hence with the number of active firms in the cartel (if entry costs are sufficiently large). Using Porter’s weekly data set on the Joint Executive Committee (JEC) from 1880 to the 16th week of 1886, the chapter does find empirical support for the hypotheses that entry increases the probability of observ­ ing a breakdown o f the cartel, and that it increases the length o f the punishment period. Interestingly, Porter (1985) found support for the latter but not for the former.

Chapter two contributes to the analysis of the nature o f the difficulty for collusion when firms differ in “size”, depending on the underlying reason which explains the dif­ ferences in firms’ sizes. In particular, the chapter analyses the conditions under which an industry-wide collusive outcome can be supported when an infinitely repeated game is played between asymmetric quantity setting firms which produce a homogeneous product. Following Perry and Porter (1985), asymmetries are dealt with by assuming that firms can have a different share o f a specific asset (say, capital) which affects marginal costs. In this context, a firm is considered “large” if it owns a large fraction of the capital stock, and “small” if it owns only a restricted proportion of the capital available in the industry. The model assumes that firms use optimal punishment strategies with a

stick and carrot

struc­ ture in the style o f the ones which have been characterized in general by Abreu (1986, 1988). In particular, Abreu’s work is extended to consider a class of “proportional penal codes” (that is, along the punishment path firms produce in proportion to their assets).

The chapter’s main results can be summarized as follows. First, it is shown that firms’ incentives to disrupt the collusive agreement crucially depend on the distribution of assets amongst firms involved in the agreement. Second, from the analysis of the

Introductory overview 4

pact o f changes in the distribution of asset holdings (due to mergers, transfers or split-offs) on the sustainability of tacit collusion, it is found that if a merger (or other asset transfer) induces a more even distribution of assets, this tends to foster collusion. These results em­ body interesting implications for practical application of competition policy. In particular, the conclusions of this chapter shed some light on the analysis of the complex problem of assessing how a merger would affect collusion possibilities, an issue widely debated in today’s competition policy (under the name of

joint dominance

, which refers to the possi­ bility that firms reach a collusive outcome after a merger). The analysis clearly suggests that asymmetries in cost functions should be taken into consideration when predictions are made regarding the facility o f collusion after an asset transfer takes place. In addition, our results confirm that, as initially stressed by Compte Jenny and Rey (1997) and more re­ cently by Kuhn and Motta (1999), a systematic analysis of market shares and concentration indexes does not always provide a reliable guide to evaluate potential effects on the level of competition in the market induced by an horizontal meiger.

Chapter three aims at providing empirically testable implications regarding the rela­ tionship between market size and concentration in

endogenous sunk cost industries

(Sutton (1991, 1998)), that is, industries where firms are involved in research and development (henceforth, R&D) activities with the aim o f enhancing the perceived quality of their prod­ ucts. Sutton (1991, 1998) has shown that in industries o f this type very fragmented out­ comes cannot arise as equilibrium outcomes in large markets. He shows that, under very general conditions, a lower bound to concentration exists and is bounded away from zero, no m atter how large the market is. However, as pointed out by Bresnahan (1992) and

Introductory overview 5

Scherer (2000), an important question left open by Sutton’s analysis is whether an upper bound to the degree o f concentration can be characterized in this type of industries. Chapter three addresses this question by extending the linear-demand model with horizontal product differentiation proposed by Sutton (1998). The chapter incorporates a post-entry additional stage into Sutton’s framework, where firms may endogenously form coalitions. By forming coalitions (meiging), firms cooperate and eliminate duplication efforts in R&D activities to enhance product quality. Hence, apart from reducing competition in the market, a m erger allows firms to realize a cost advantage over the unmerged rivals.

A novel feature o f this chapter is that it employs a coalitional stability concept which assumes that firms are endowed with foresight, in the sense that when making merger decisions, they look ahead and anticipate the ultimate outcome of their actions. This chapter is, therefore, related to a relatively new strand of the literature on farsighted stability (see, for instance, Chwe (1994), Xue (1998) and Diamantoudi andX ue (2001)).

The analysis leads to the following two main conclusions. First, independently of the size of the market, arbitrarily concentrated outcomes can arise in equilibrium. Therefore, this chapter shows that in

endogenous sunk cost industries

, an upper bound to concentration exists and is independent of the size of the market. Interestingly, in some equilibria in which a merger to monopoly is the unique equilibrium outcome, it turns out that firms belonging to the monopoly ‘grand’ coalition earn strictly positive profits even under the threat of entry. Second, it is also shown that if products are sufficiently good substitutes (or, if investment in R&D is sufficiently effective), duopoly coalition structures can only arise in equilibrium

Introductory overview

H U L L

6

if composed of sufficiently size asymmetric coalitions. The results, therefore, complement those o f Sutton (1991,1998).

Finally, as a practical remark, it should be stressed that all chapters o f this thesis are self-contained and, therefore, can be read independently from each other as an independent article. This has the advantage that readers who are interested in only a part o f the work presented here can gain easy access to their point o f interest.

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Chapter 1

Entry Effects on Cartel Stability and the Joint

Executive Committee

1.1

Introduction

The objective of this chapter is twofold. From a theoretical perspective, the chapter con­ tributes to the analysis of entry effects on cartel stability under demand uncertainty. From a more applied perspective, we empirically test some theoretical predictions about how entry can affect the pattern o f collusive behavior o f a group o f firms organized in a cartel agree­ ment to coordinate prices, making use o f data on the US railroad cartel of the tum-of-the century.

We develop an extended version o f the model proposed by Green and Porter (1984). In their seminal article, Green and Porter analyze infinitely repeated oligopoly games where market demand is subject to exogenous shocks and the firm’s (past) actions are not observ­ able, but they do not consider the possibility of entry. We thus reexamine their model to understand how the stability of the collusive price structure can be influenced by an in­ crease in the number of firms in the agreement or by the existence of a pool of potential competitors.

The framework we use in order to explicitly model the entry process is related to Harrington (1989). We study two types o f collusive equilibria in a repeated Bertrand game (with random demand and unobservable prices) between a set o f active firms and a set of

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1.1 Introduction

potential competitors that can enter the market by paying a one-time, fixed, sunk cost (entry cost). In a collusive equilibrium, the initially active firms have no incentive to cut prices in non-reversionary periods (because this would trigger a “price war”) and the potential entrants have no incentive to enter, because the present value o f the expected profits from entering the cartel is not sufficient to cover the entry cost. In the first type of equilibrium it is expected that entry would trigger a price war. In the second type of equilibrium it is expected that entry would be accommodated with a more inclusive agreement. It should be noted, however, that a major difference exists between the model developed in this chapter and H arrington’s (1989) framework. While Harrington’s results are obtained in a context of alm ost perfect information,1 our analysis restricts the information available to firms, in the sense that at each period o f time, apart from past entry decisions a firm knows only its own past prices and output levels. We find that entry does reduce the scope o f collusion under both types of equilibria. In addition, in contrast with what the previous literature has pointed out with respect to the experience of the US railroad cartel, it is shown that, from an

ex-ante

point of view, the existence of a pool of competitors is a more important constraint on the maintenance of a stable agreement when a potential entrant expects to be accom m odated by incumbents if entry occurs. We then look for empirical evidence on the role played by entry using the data set from the experience of the Joint Executive Committee (henceforth JEC), a pre-Sherman Act (legal) cartel.

T he Green and Porter (1984) model has been subject to previous empirical tests us­ ing the JEC data set. The model suggests that in industries working in a context of

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1.1 Introduction 9

feet observability, price patterns include shifts between collusive regimes and competitive regimes along the collusive equilibrium path. Porter (1983b) and Ellison (1994), among others, demonstrated the existence of such regime shifts while examining this railroad car­ tel.2 In another paper, which is probably the closest to the empirical application developed in this study, Porter (1985), still using the JEC data set, analyzes empirically the determi­ nants of both frequency and duration of competitive reversions of finite length. The main goal of the econometric work developed by Porter was to determine whether the JEC was less successful in maintaining cooperative prices because of entry of new firms. To this end, he ran regressions for both the full sample and two subsets of the overall sample in which there were structural changes due to entry occurrence, to test for the impact of entry on the likelihood of a price war beginning. As noted by Porter himself, using the incidence of competitive episodes reported by the press at the time as the dependent variable, the re­ sults obtained are quite discouraging. Using the same data set but considering a different specification of the econometric model and creating new variables to explore what causes price wars to occur, our findings confirm the predictions of the reexamined version of the Green and Porter (1984) model presented in this chapter. In particular, unlike Porter, we find that a larger number of firms in the industry increases the percentage o f periods spent in a price war. Two different forces justify this result. First, the higher number of firms in the agreement is, the higher the (one-shot) incentives to disrupt the collusive agreement are. Second, as is shown theoretically and contrary to what Porter (1985) claims,3 the

opti-2 However, while Porter (1983b) obtained the result that firms’ price cost mark-ups were consistent with a Cournot behavior, Ellison (1994), allowing for serial correlation in the demand between periods, came to the conclusion that cartel members were setting prices collusively between price wars.

3 See Porter (1985, p. 419).

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1.2 The basic model

10

mal punishm ent length is always increasing with the number of firms in the basic version of the m odel and increases as well with the number of active firms when entry is explicitly m odeled, as long as the entry (sunk) costs are sufficiently high.

The chapter is organized as follows. Section 1.2 includes a brief description o f the structural features of the basic theoretical model and some preliminary predictions about entry effects on cartel stability. The structure presented is based entirely upon the model developed in Green and Porter (1984). Departures from their assumptions are noted be­ low. In Section 1.3 we develop the central analytical argument, extending the analysis o f the preceding section to consider the possibility of entry of new firms. Section 1.4 briefly reviews the operations of the JEC. A description o f the data used in the empirical applica­ tion and the estimation results are provided in sections 1.5 and 1.6, respectively. Section

1.7 contains some concluding comments.

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1.2

The basic model

Following Tirole (1988), we will develop a model in the spirit of Green and Porter (1984) in w hich a cartel is sustained by oligopolistic firms acting noncooperatively in a context o f demand uncertainty.4 In this section we briefly remind the reader of the main features of this well-know n model for the case where entry is not allowed. In the following Section

1.3, w e extend the model to analyze entry.

4 We depart from Tirole’s approach by considering that there are n firms in the agreement rather than two and also by allowing for a wider range of possible prices along the collusive path.

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1.2 The basic model 11

There exist n firms producing a homogeneous good and facing the same unit cost c. Firms choose prices in every period. Demand fluctuates randomly and its realizations are assumed to be independent and identically distributed (i.i.d.) over time. In each period there are two possible states of nature. With probability a the demand is zero (“low-demand state”) and there is a positive demand with probability (1 — a ) (the “high-demand state”). In the latter case, demand is split into equal parts corresponding to those firms charging the lowest price.

Firms do not observe their rivals’ prices. Thus, from the point of view o f each single firm in a cartel, a low demand for its product may be due to either secret price cutting by some competitors or a bad market demand shock.

A strategy, that is, a contingent plan of action, for a firm

i

in the repeated game is an infinite sequence 5 t- = (5?,

S } ,

S J ,...), where Sj* €

R+

is a determinate initial price level, and

Sf

:

(R+Y -+ R+

is a function that maps the prices charged and quantities faced by firm

i

in periods 1,2 ,...,

t —

1 into a price p-, for firm

i

in period

t.

In this game, we look for a Nash equilibrium with the strategy for the i - t h player defined in the following way:

P

.

(

1

.

1

)

.... W“1^ - 1))

p*

if q

I

1 > 0, pi' 1 = p * , or Vr € [f -

T ,t -

1],

p j

= p c,

qj >

0. p c otherwise ( 1 . 2)

where i = 1,2,..., ( ( g ° , p ° ) (i Ì“ 1^ * “ 1)) *s the partial history with length

t

observed by firm

p* e

(pc, Pm] is the collusive price, while

pc

and

pm

represent the competitive and the monopoly price, respectively.

1.2 The basic model

12

In words, the game lasts forever and initially each firm charges the collusive price. As soon as one o f the

n

firms in the cartel observes a zero demand (and, therefore, earns zero profits), a punishment phase o f

T

periods is triggered in which every firm adopts a Bertrand behavior.5 At the end of the reversionary episode all firms return to collusive behavior and share the collusive profit (II*) until a zero demand is again observed by some (or all) firms.6

Note that the length of the optimal punishment period can be neither zero nor infinite. L e t

V*

represent the present discounted value o f a firm ’s profit from date

t

on, as­ suming that date

t

belongs to a collusive phase. Analogously, let

V~

denote the present discounted value of a firm’s profit from date

t

on, assuming that date

t

is the beginning o f a punishment period.7 In this context w e have:

Equation (1.3) says that with probability (1 —

a)

the demand state is high, each firm earns its share of the collusive profit and the game remains in the collusive phase, and thus each firm has the valuation V^+ . However, with probability a , the “low-demand state” is achieved and in the next period a punishment phase starts. Equation (1.4) gives the present

5 As was pointed out by Fudenberg and Tirole (1991), “no player can gain by deviating in the punishment phase, since play there is a fixed number of repetitions of a static equilibrium.” (p. 186).

6 In the light of Abreu et at (1986), we know that it is possible that a global optimum might not be achieved with Nash reversion. However, since the competitive price Nash equilibrium achieves the minmax payoff profile, there is no room for a stronger punishment than Nash reversion in this setting.

7 Because of stationarity, neither V * and V~ depend on time. In addition, the subscripts denote the number of firms in the agreement.

and,

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1.2 The basic model 13

discounted value of the expected stream o f profits at the beginning o f the punishment phase phase.

By solving the system of equations (1.3) and (1.4) one obtains:

V*

1—(1

- a ) S - a S 1+T

V r = „ , n * (i - “ ) — .

_n

(1.5) M , n *

n

(1.6) " 1 — (1 — a )

6

— a £ 1+r

As we have seen, cartel members’ strategies (1.1) and (1.2) prescribe a mechanism to punish deviations from the price structure agreed upon. Whether this punishment mech­ anism is self-enforcing will depend on the trade-off between potential short-run gains from deviation and the present value of expected future losses. This trade-off is captured by the analysis of the incentive compatibility constraint:

i • i :i

V* >

(1 -

a)

(IT +

SV-)

+ a ( iV " ) , (1.7)

where

6

€ (0,1) is the common discount factor. The right-hand side o f the inequality shows that if the demand is high and the firm decides to undercut, it earns all the one-shot collusive profit, but by deviating it will trigger a punishment reversal in the following

T

periods. If the demand is zero, it will earn zero profits in the current period and a punishment phase starts in the next period.

Using equation (1.3) we can rewrite the incentive compatibility constraint in the fol­ lowing way:

s

(K+ -

v~)

> n* - 51.

(i.8)

b r u s i i « » - * I I U I X ^

1.2 The basic model 14

This relation says that a prospective deviant will decide to respect the agreement if the present discounted value o f the long-run net gain from collusion is greater than the short-run gain from deviation.

T h e next proposition derives the optimal length of the punishment period

T*

as a function o f the parameters in the model and shows that it is increasing in the number o f firms belonging to the agreement. The intuition which underlies this result is simple. On the one hand, when more firms belong to the collusive agreement, the gain from secret price cutting becom es larger as the deviating firm can obtain m ore share. On the other hand, each firm ’s continuation payoff in the collusive equilibrium decreases, which reduces the impact o f any finite-period punishment. Combining these two effects, one concludes that the optim al length of the punishment has to increase in order to maintain the incentive for the firm s to stick to the collusive price.

P ro p o sitio n 1

I f the number o f firms is sufficiently low, i.e.,n

< 1 /(1 — <5(1

— a)), the

optimal punishment length T* is determined by

T * = — In n ( 1 - g ( 1 ~ Q ) ) ~ 1

ln 5

6 (an —

1)

where T* is an increasing function o f the number o f firms in the agreement.

Proof, Substituting (1.5) and (1.6) into (1.8), some algebra shows that, in order for the incentive compatibility constraint to hold, one must have that:

rm linn

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1.3 Considering entry 15

where we assume

a <

1 /n in order for the r.h.s. of eq. (1.9) to increase with

T.

As Tirole (1988) and others have mentioned, the optimal punishment length

T*

should be chosen to maximize the discounted joint profits and, therefore, should be the minimal

T

for which eq. (1.8) (or, equivalently, (1.9)) holds. Rewriting (1.9), one obtains:

T ^ n — 1 — ¿n (1 — a )

8 1 <

8

( a n — 1) (

1

.

10

)

which in turn implies that

- l n < 5 ¿ ( a n -1) ’ (

1

.

11

)

where, since a < 1/ n , one must have that n < 1/ (1 —

8

(1 — a ) ) in order for

T*

to exist. Now, taking the derivative o f

T

* with respect to the number o f firms, one obtains:

dT*

(1 - a) (1 -8 )

(U 2 )

dn

In

8

(o n — 1) (n (1 —

8

(1 — a )) — 1 )7

which is always positive given that, as already mentioned, o < 1/ n and n < l / ( l - 6( l — a )). This completes the proof. ■

First, it should be stressed that an optimal punishment length only exists if the number of firms in the agreement is sufficiently low. Otherwise, the cartel is never (internally) stable. Second, and most importantly, notice that this result is in sharp contrast with what Porter (1985) indicates as a possibility, i.e., that as the number o f firms increases, “the cartel may want to employ shorter, and so more forgiving, punishments when reversions occur, in order to partially offset the increased fraction of time spent at competitive prices.” (p. 419). This result will play a central role in the explanation of the main hypothesis tested in the empirical section of this chapter, namely that cartel stability is negatively correlated with the number of firms in the cartel agreement.

1.3 Considering entry

16

1.3

Considering entry

We will now describe a new version of the Green and Porter (1984) model in which th e entry process is modeled explicitly. We will study a particular subset of the set o f Perfect Bayesian Equilibria. In particular, we will assume that once entry occurs, incumbent firm s can either engage in aggressive behavior for a while (Case 1) or accommodate the new entrants (Case 2).8 Throughout the analysis o f each o f these cases, we will first give th e necessary and sufficient conditions for a non-trivial degree o f collusion to be sustainable and characterize the optimal length o f the punishment period which is required by th e equilibrium conditions. We then explore the stability issue in further detail, by analyzing the determinants of the proportion of periods spent by the cartel in price wars.

1.3.1

The new model structure

Assume a countably infinite number o f active and inactive firms represented by the set Z.

A 1

C Z , where the inclusion is

strict

, denotes the set o f active firms in period

t

and ¡A*] = TV* (where |A*| is the number o f elements o f A*).9 Hence, Z — A* represents the set of potential entrants in each period

t.

Further, let both active and inactive firms have th e same marginal costs of production.

8 We are not considering here the situation where incumbent firms coordinate their behavior to force an entrant to exit from the industry. During the operation of the JEC, it was common knowledge that new competitors faced a “no-exit constraint”. As a consequence, “it would not be rational for a railroad cartel to engage in predatory pricing practices in response to entry.” (Porter 1985, p. 420). Thus, we have decided not to cover predatory pricing here, since it is clear that the use of this kind o f threat against potential entrants will not have been credibly carried out by the railroad firms belonging to the nineteenth-century railroad cartel, whose pricing strategies we will discuss in the applied part of this chapter.

rtiriiiamitkii iliihM iiU

1.3 Considering entry 17

We also assume that, for a potential entrant, entry and price decisions are not simul­ taneous. At each period

t,

the Ari active firms simultaneously announce the price to charge (p- denotes the 2-th firm’s price in period i). At the same time, potential entrants decide about entry. A one-time entry (sunk) cost

K

(where

K

> 0) has to be incurred if entry takes place. It allows the firm to begin production one period later. Hence, prices are a post-entry decision for a potential competitor and past entry decisions are assumed to be perfectly observed by all active firms.10

If a firm

i

is initially active (

i

€ A0), its overall strategy - S* - can be represented as an infinite sequence o f action functions (one for each period)

Si =

(£?,

S } ,

5* ,...), where Sf €

R+

represents the initial price charged by firm

i

and

Sj

: (/?+ x 2Z)* —►

R+

. The domain of an action function 5J is the Cartesian product between the set of feasible prices

pj,

the set o f possible outputs11

qj

and the set o f active firms in period r , where r 6 {0, ...,f — 1}. The range of 5* is represented by the set of possible prices that firm

i

can charge in period

t.

Hence, period

t

action function o f an arbitrary active firm

i

tells it which price to set in the

t —

th period as a function of the feasible histories observed over the periods {0,

t

— 1}.

More formally:

where A0) , (g}” 1,/?!- 1 , Ai-1) ) is the partial history with length

t

observed by firm i, which is denoted by

h\

G

H\.

Notice that

h\

can be partitioned into the public partial 10 Entry is generally a time-consuming process; therefore, following Harrington (1989), we assume that

incumbent firms are able to change their price decisions in response to entry.

1.3 Considering entry 18

history (>4°,...,

A*"1)

and the private partial history ((g ° ,p ° ),..., (g*-1 *?!-1 )) observed b y firmi.

I f firm

i

is instead initially inactive (i 6

(Z

— A 0)), then its overall strategy is

Ei

= (£{\

E

} , ...,

E l

, ...). In the first period o f the game (f = 0),

E f

€ {Out, In} x {oo}, w here Out means “Do Not Enter the Market” and In means “Enter the Market” . Moreover, w ith respect to the price decision, the set o f feasible prices is a singleton.12 For each period

t €

{ 1 ,2 ,...} , there exists an action function

E\,

which, given the history observed up to period

t —

1, tells the firm whether or not to enter at the beginning of period

(t

-f 1). W e consider entry as an irreversible decision. Thus, if entry occurs, firm i ’s strategy specifies the prices to be charged for the remainder of the horizon. The observed (partial) history is composed by the own price and entry decisions and by the demand faced by the firm up to the previous period. In formal terms,

E\ :H\-+

{Out, In} x

{R+

U {oo}}, w here

HjC

({Out, In} x

R \ ) .

Moreover,

V

h\

€

Hi,

if (...,

{xjypjj qj

) , ...), where

x j =

Out, then g[ +1 = 0.

It should be also noted that:

El

((Out, o o ,0 ) ,..., (Out, oc,0)) € {Out, In} x {oo} ,

while

El

((Out, o c , 0 ) , (In, o o ,0)) € {In} x

R+.

12 The only feasible price is p® = oo, thus the firm will not face a positive demand in the first period of the game.

B

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1.3 Considering entry 19

In the discussion that follows, we will identify, for each o f the considered cases of post-entry reaction, the conditions which should be satisfied in order for the incumbent firms to sustain a non-trivial degree of market power without giving rise to either internal or external defection. Following Harrington (1991), we consider that internal defection takes place when some active firm secretly undercuts the price while external defection occurs if some potential competitor decides to enter the m ark et13

1.3.2

Case 1. Cartel Breakdown

In this case we consider a situation in which incumbent firms respond to entry by reverting to competitive pricing for a finite length of time

(T

periods). Since we want to understand how incumbent firms effectively sustain collusion, avoiding not only internal deviations for profit, but also entry of new competitors, an active firm’s strategy is designed in the following way:

S ? = p \

if

q j'1

> 0,

pl~l — p *

and

A

* =

A l~l ,

or V r 6

[t

- - 1] ,

p j = pc, q j>

0 and

A T

=

Ar~l

otherwise (1.13) (1-14) where

t —

1,2, . . . ,

i

£

A °t N° =

|v4°| =

n.

Hence, the incumbent firms charge the collusive price (p*) at the beginning o f the game and continue to set it as long as no firm faces a zero demand and no entry occurs. However, if some firm (active or inactive) has deviated from the proposed path or if a

13 Note that the basic framework of Section 1.2 can be viewed as a special case of the extended model we present in this section, where the entry sunk cost is prohibitive (K —+ oo) and, therefore, external stability is not an issue.

■<

20

1.3 Considering entry

low-demand state has occurred, then the collusive price is reestablished after a (temporary) punishment phase.

O ur aim is to find a trigger strategy equilibrium such that strategies (1.13), (1.14) are optimal and there is no incentive for the entry of new firms into the industry. To this end, consider the following strategy for a potential entrant:

E f

= Out

E!

Out

Si

if

i e Z - A l

if

i

€

A 1

(1.15) (1.16) where

t

= 1, 2, . . . ,

i € Z — A 0.

This strategy calls for a potential entrant not to enter the industry. However, if th e firm decides to enter, then the strategy also prescribes the price conduct which it should follow after its entry.14 Remember that the incumbents will react to entry by setting a price

pc

during a finite punishment period. Hence, once inside the market, the best response o f the entrant is to set

pc

during the punishment period which is triggered by its entry, since the Bertrand solution constitutes a N ash equilibrium for the one-shot price game which is played in each single-period, given the number of active firms in the industry.

A free entry trigger strategy equilibrium is defined as a triple

(p*,

Tj, such that the strategies in (1.13)-(1.16) form a Perfect Bayesian Equilibrium (PBE). The following two conditions are necessary and sufficient for (1.13)-(1.16) to form a PBE:15

s {Y+

-

V - ) >

n * - 2 1 (1.17) 14 It should follow the strategy of an initially active firm (strategy (1.14)) for the remainder of the horizon. 15 Notice that subgame perfection cannot be used as the equilibrium concept. This is a dynamic game with

unobservable actions in which the only proper subgame is the whole game itself. Moreover, active firms are able to observe actions which are off the equilibrium path (remember that they know the set of active firms in the past periods).

1.3 Considering entry

21

and,

6V~+ l- K <

0. (1.18)

Notice that not only the collusive price, but also the length o f the punishment are chosen (optimally) by the cartel to maximize the expected discounted joint profit subject to the constraints (1.17) and (1.18). As can be easily verified from equation (1.5), Vn+ is a decreasing function o f

T.

As in Section 2, expression (1.17) represents the incentive compatibility constraint. In Proposition 1, it has been shown that this constraint holds as long as the punishment length is large enough. In particular, if

T

> T \ where T* is given by eq. (1.11), then each of the

n*

active firms finds it optimal to go along with the collusive path and to charge the collusive price

p*

since the discounted loss from cheating is greater than the one-shot gain from deviation. On the other hand, condition (1.18) makes further entry into the industry unprofitable. Thus, when this latter condition holds, existing profits might be positive because profits with further entry16 are expected to be negative, which means that “Out” is, in fact, a best response for an inactive firm.17

In what follows, the optimal punishment length in a cartel breakdown scenario is derived and characterized.

Proposition 2

Let T* be as in Proposition 1

.

In a scenario in which incumbent firms

respond to entry by reverting to a price war during a finite length o f time, and assuming

16 Defined net o f the (sunk) costs of entry.

17 The condition in (1.18) can be interpreted as a violation of the participation constraint corresponding to the pool of potential competitors.

LvjLf lib -U »L/ 11 Lij: ■t • ^ rilN M M M lU ib h b É tlU U

1.3 Considering entry

22

0 <

K

< (1 1 7(71* + 1)) (1/(1

—8)), the optimal punishment length T\ is given by Tj

= m ax {T*,

Tpci}, where

1

K

(n* + 1 -

8

(1 -

a)

(ra* + 1)) ^ I n i ¿ ( ( l - a j n ' + t f a i n ’ + l) ) '

Proof. From Proposition 1, we know that the incentive compatibility constraint (1.17) holds if and only if

T

>

T*,

where

T*

is defined by eq. (1.11). Now, using eq. (1.6), it is straightforward to show that:18

6r

V" ,+1 1—(1 —a ) < 5 - a i 1 + r (1 Q) I P

n* - f 1 ( U 9 )

Hence, substituting (1.19) into eq. (1.18), some algebra shows that incumbent firms will be able to coordinate their price strategies without giving rise to entry of new competitors if and only if:

^ r'+ i <

K

(n * + 1) (1 ~ (1 ~ <*) Æ)

(

1

.

20

)

(1 — a ) II* +

K a

(tz* 4* 1)

T h e l.h.s. o f condition (1.20) is easily seen to be decreasing in

T'.

Notice also that when

K

> ( p^ ) ,19 the r.h.s. o f condition (1.20) is greater than or equal to one and, therefore, condition (1.20) is trivially satisfied. Let us, therefore, consider in what follows the case in which

K

< — ^ (tt$)- If this is the case, some algebra shows that condition (1.18) holds if and only if

> J _ In

K

(re* + 1 ~

6

i1 ~ tt> fo* + ^ =

T

In 8

ó ( ( l - a ) n * + t f a ( n . + l)) “ pcl‘

18 We here obviously assume that the length of a price war depends on the number of active firms in the agreement. Hence, given that V~.+1 is computed for a case in which there are n* + 1 firms in the agreement, notice that we are now considering a general value T ' (different from the one used in V~., T).

19 When this is the case, entry (sunk) costs are extremely high (say, prohibitive) and, therefore, external stability is not an issue.

MMMMlttÉlliUÜti ü m u M u » M ^ U U M u w u y > . - u M . ü w U U „ . , u ^ , _t J .

1.3 Considering entry 23

Hence, the minimum length o f the punishment period for which both the incentive and the participation constraints (eqs. (1.17) and (1.18)) hold is given by

f \

= m ax {T% Xpd}. ■

Two important remarks are in order at this point. First, using (1.21) it can be easily shown that

dTpci

1____________ IT (1 -

a)

___________

dn*

In

6

(n* + 1) ((1 —

a)

II* +

K a

(n* + 1)) ’

( 1.22)

and

dT,

pci 1

n* (1

-

a)

< 0. (1.23)

d K

In

6

(II* (1 - o r ) +

K a

(n* + 1))

K

It should be stressed at this point that the

initial

number o f firms

n*

is a parameter of the model affecting the viability of collusive agreements. In equilibrium, the number of firms does not change, and so it makes sense to do comparative statics with respect to

n*.

Let us now turn to the interpretation o f the previous results. From (1.22), it can be concluded that the greater the number of firms in the agreement or the higher the height o f the entry barriers is, the lower will be the length of the punishment period which must exist in order to avoid entry and allow incumbents to sustain a non-trivial degree of cooperation. Second, eq. (1.23) calls attention to the fact that for sufficiently high values o f the entry (sunk) cost, condition (1.18) is non-binding. In particular, and as already mentioned in the proof of Proposition 2, if

K >

II /( n + 1 ) ( 1 / (1 — 6)),20 then (1.18) is trivially satisfied. In the next Lemma, however, tighter predictions are given regarding the ranking between the two relevant thresholds for the punishment length,

T*

and Tpcl.

20 That is, the value of the entry cost is higher than or equal to the present value of individual shares in the collusive profits in case no punishment is triggered in the future periods.

i U A >

1.3 Considering entry 24

L e m m a 1

Let T\, T ' and Tpci be as in Proposition 2

.

T\

=

T* if and only if K > K\>

where

K

n* (1 — n* (1 — 5 (1 — q)))

1

n*

+ 1 1 —

6

Proof. M aking use o f ( l.ll) a n d ( 1 .2 1 ) , some algebra shows that

T* > Tpci

if and only if

(1 - n* (1 -

6

(1 - a ) ) ) ((1 -

a)

I P +

K a (n*

+ 1 ) ) (1 —

a n

*)

K (n*

+ 1) (1 —

S

+ ¿ a ) “ which in turn implies that

(1.24)

n * ( l - n » ( l - g ( l - o ) ) )

"

n*

+ 1 1 - 5

= K

l

Hence, when

K

>

K\, T\

= m ax {

T

*,

Tpa}

=

T*.

■

(1.25)

In words, Lemma 1 shows that fo r sufficiently high values of the entry (sunk) cost

K>

the incentive compatibility constraint is the one that is binding and, therefore, the optimal length o f the punishment is given by the minimum value of

T

for which this incentive constraint is satisfied,

T\

—

T

*. As the next Corollary shows, this result is important to understand w hat is, in this case of the extended version of the model, the impact of entry on the duration o f the optimal length o f the punishment phase.

C orollary 1

Let K \ be as in Lemma 1. The optimal length o f the punishment in a cartel

breakdown scenario

, Ti,

increases with the number offirms if and only if K

>

K\.

Proof. If

K \ < K <

(II*/(n* + 1 ))(1 /(1 — 5)), then applying Proposition 2 and Lemma 1, one concludes that

T\ = T*.

Now, from (1.12) we know that |^ r > 0. If, instead,

K >

+ 1)) (1/(1 — 5)), then entry costs are prohibitive, the incentive

éumttìàM

1-3 Considering entry 25

compatibility constraint is the one which is binding and, as already mentioned, from (1.12) we know that §£7 > 0. Lastly, if 0 <

K < K u

then from Lemma 1, one has that Ta =

T

pcj and from (1.22) we know that < 0. M

So, even when potential entrants are threatened with a subsequent (temporary) cartel breakdown if entry takes place, strictly positive entry sunk costs must exist in order to discourage entry into the industry. In addition, if sunk entry costs are sufficiently high, then the optimal punishment length chosen by cartel members is increasing in the number of firms belonging to the collusive agreement.

1.3.3

Case 2. Accommodation

Rather than following the policy of reacting to entry by starting a finite punishment period, cartel members might decide to accommodate the entrant by achieving a new collusive outcome.21 If a potential entrant anticipates that, entering at period t, the

N 1,

active firms will adopt post-entry accommodation behavior, then the necessary and sufficient conditions for a triple (p*,X2,

n*

) to constitute a free entry trigger strategy equilibrium are now given by:

6(Vn+.- V - . ) > T l ' - ^

(1.26)

and,

(1.27)

1.3 Considering enùy

26

W henever condition (1.26) holds, an individual active firm predicts that it will make more profit by being loyal to the cartel agreement than by being disloyal; therefore, the agreement is unlikely to breakdown because of an internal deviation.22

Condition (1.27) specifies that, even though incumbents will allow new competitors to join the collusive process, potential entrants anticipate a negative post-entry profit, which means that “O ut” is an optimal choice for them.23

A s in the previous section, we can now derive the optimal punishment length in a context where entering firms are accommodated by incumbents in a more inclusive agree­ ment.

■1 »

P rop o sitio n 3

Let T

*

be as in Proposition 1. In a scenario in which incumbent firms re­

spond

to

entry

by

accommodating

the

new

entrants,

and

assuming

(n * /(n * + 1))

(S

(1 - a ) / (1 -

6

(1 - a ) ) ) <

K

< ( n '/ ( n * + 1)) (5/(1 -

6)), the op­

timal punishment length

f 2

is given by

f2 = m ax

{ T 't Tpc

j},

where

rn

1 , A r(n- + l ) ( l - 5 ( l - Q) ) - 5 I P ( l - a )

***

ln5

KSa(n' + 1)

Proof. From Proposition 1, w e know that the incentive compatibility constraint (1.26) holds if and only if

T > T*f

where

T*

is defined by eq. (1.11). Now, from (1.5) one can

22 It should be noted that although incumbent firms follow a policy of post-entry accommodation, we are still assuming that internal defection is followed by a finite period of cartel breakdown. Hence, conditions (1.17) and (1.26) coincide.

23 Again, both p * and X2 are chosen (by the cartel) to maximize the discounted profits, but now subject to constraints (1.26) and (1.27).

1.3 Considering entry

27

easily conclude that

^

l - ( l - a U - a < 5 1+T" (1

a ) n* + l

n*

( l - a )

(1.28)

Substituting now (1.28) into eq. (1.27), some algebra shows that condition (1.27) is satisfied if and only if:

1 + r, ^

K ( n '

+ 1) (1 - (1 - a )

8) - 6

(1 -

a)

II*

K

(n* + 1) a (1.29)

where we have to assume two restrictions regarding values which the entry sunk cost can take. First, we assume that

K >

m

order for the r.h.s. o f eq. (1.29) to be positive. Second, we suppose

K

< (—j ) to rule out the case in which the r.h.s. o f eq. (1.29) is greater than or equal to one and, therefore, the previous condition is trivially satisfied. Now, since the l.h.s. of eq. (1.29) is easily seen to decrease in T ", some algebra shows that condition (1.29) (and, hence, (1.27)) holds if and only if:

r a

b * <"■+» s - ƒ P : ”» -«El1

- s i

, !•„.

(,.30)

In

6

K oa (n*

+ 1 )

Hence, the minimum length o f the punishment period for which both the incentive and the participation constraints (eqs. (1.26) and (1.27)) hold is given by 7*2 = m ax{r*,T pc2}. ■

Notice that now, contrary to what has been found for the cartel breakdown scenario, in order for

T

p c 2 to exist, the entry (sunk) cost has to be sufficiently high (and not only positive).

As in Case 1, the analysis will now focus upon the sensitivity o f the critical level for the punishment length for which the participation constraint (1.27) holds,

Tpc

2, to changes in the number o f firms in the agreement or in the level of the entry cost. Making use o f

1.3 Considering entry

28

(1.30), it is straightforward to show that:24

d T p c 2

_ _______________ IP (1 -

a )

________________

6 _

d n

• ~ (n* + 1)

(K (n’

+ 1) (1 — ¿>(1 -

a)) —

OT* (1 — a )) I n i < U’ (1.31) and

9T ,c

2

________________ n* (1 - q )_________________

6_

d K

K { K ( n

* + 1) (1 -

6(1

- o )) - <511

*

(1 - a)))ln<5 (1.32) H ence, as in Case 1, the higher the values o f

n*

and 7f, the lower the minimal length of the punishment period which is compatible with an anticipated unprofitable entry. A s before, one can now show that, since

Tpc

2 is decreasing in i f , for sufficiently high levels of the entry sunk cost, condition (1.27) is a non-binding constraint. The next L em m a formalizes this result.

L em m a 2

Let T2> T* and T

pc2

be as in Proposition 3. T2 = T* if and only if K >

i f 2,

where

K 2

I P <5(1- a r c » )

n*

+ 1 1 — <5

Proof. From (1.11) and (1.30), some algebra shows that

T*

>

T

pc2 if and only if: i f a (n* 4- 1) (

n

* — 1 —

Sn

* (1 — a ))

(on* — 1) ( i f (1 — (1 — a ) <5) (rc* + 1) — ¿ IP (1 — a )) which in turn implies that25

< 1, (1.33)

i f

>

IT <5 (1 -

an*)

=

rc* + 1 1 — 5 (1.34)

Hence, if i f

>

i f 2,

T2

= max {T*,Xpc2} =

T*.

24 Remember that, as was mentioned in the proof o f Proposition 3, we assume that K > Otherwise, there is no (finite) optimal punishment length compatible with cartel external stability. 25 Remember that a < 1/n*.

1.3 Considering entry 29

•tuL ..

-C orollary 2 Lei K2

be as in Lemma

2.

The optimal length of the punishment in an

accommodation of the new entrants scenario

, ¿2,

increases with the number o f firms if and

only if K

>

Proof. If jFsT >

K

2, then applying Lemma 2, one has that T2 = T"\ Then, from (1.12)

we know that |£ r > 0. If, instead, - ^ <

K

< /C2, then by Proposition 3 and Lem m a 2, one has that

T

2

= Tjx

2 and from (1.31) we know that < 0. Lastly, if

K

< 1ig1(^ia)» ^ already mentioned in the proof of Proposition 3, the cartel is not externally stable for any (finite) value of the punishment length. ■

t ;

. «

Before closing this section, let us compare the two critical levels for the entry cost derived in Lemma 1 and 2. Using (1.25) and (1.34), simple algebra shows that:

K 2 - K i =

-7 7 -7 (n* - 1 ) > 0. (1.35)

n*

+ 1

This result stresses the fact that, for a given market structure (rc*), in order for the par­ ticipation constraint to be a non-binding constraint, the height of the entry barriers should be higher in the second case (accommodation o f the new entrants) than in the first one (car­ tel breakdown). In other words, the existence of a pool o f potential competitors is a more important constraint on the maintenance of a stable agreement when cartel members decide to accommodate the entrants. The intuition behind this result is just that the anticipation of a tougher price competition in Case 1 makes entry less attractive. This finding is particu-larly relevant to us since, even though it is

ex-ante

optimal for cartel members to threaten entrants with a breaking up of the cartel if entry occurs,26 accommodating the new railroad 26 Notice, however, that we are not considering the possibility of "renegotiation” of equilibria. This is

certainly an important further development of this study, since, as was stressed by Fudenberg and Tirole

imiRmuumuuu

_{JUftimi lllHIIIIUNW!*tmuiwiui’it.uiu}

_{m m r n m m}

I.

Sili i;!i

1.3 Considering entry

30

firms by allocating them market shares was considered by some authors as the incum bents’ best reply to entry during the operation o f the JEC.* 27

1.3.4

Stability discussion

The purpose of this section is to pin down what are, in the extended version of the m o d el, the theoretical predictions regarding the primary question addressed in the empirical p a rt of this chapter - understanding how entry o f new firms affects the firms* collusive p ricin g behavior.

L et

Wt

be the indicator function that takes value 1 if a price war occurs at period

t .

In addition, let P r

(\Vt

= 1) denote the stationary probability o f a price w ar occurrence in period

t.

In the next Proposition, the percentage of periods spent in a price war is derived and shown to be increasing in the punishment length.

P roposition 4

The percentage o f periods spent in a price war is given by

p'<H''-l>-T

T S

-where

P r

i}Vt

= 1)

increases with the length o f the punishment phase T.

Proof. Within this framework, along the equilibrium path and whatever the assum ed incumbents post-entry reaction, there is no cheating and/or entry o f new firms. All p ric e wars are induced by low demand shocks. The percentage of periods spent in a price w ar is

(1991), if “players have the opportunity to negotiate anew at the beginning o f each period, then equilibria th a t enforce good outcomes by the threat that deviations will trigger a punishment equilibrium may be suspect, a s a player might deviate and then propose abandoning the punishment equilibrium for another equilibrium in which all players are better off.” (p. 175).

1.3 Considering entry 31

ju st the stationary probability of being in a price war P r (

Wt

= 1). If there is a price w ar at period i — 1, then the probability that

t —

1 was not the last period of the price war is given by

(T

— 1

)/T .

P r(

Wt

= 1| IVt_j = 1) =

(T

— 1

)/T .

If, instead, there is no price w ar at period t — 1, then the probability of a price war occurrence in the following period

t

is P r(H /t = 1|

W

t-1 = 0) — a . Therefore, the unconditional probability of being in a price w ar in period

t

is defined by

Pr (W, = 1) = Pr (WU, = 1)

+ P r W - i = 0) a.

(1.36)

Now, since P r (

Wt

= 1) = P r

(W

t-1 = 1) as it is the stationary probability, one can rewrite eq. (1.36) as follows:

P r (Wt = 1) = Pr (Wt = 1)

+ [1 - Pr (W t = 1)] a.

(1.37)

Solving eq. (1.37) with respect to P r (

Wt

= 1), one obtains that:

PrW = 1) = TSr-

(1-38)

Carrying out now a simple exercise of comparative statics using (1.38), it is straight­ forward to show that:

dPr(VKt = 1)

a

d T

~ (1 + q

T )2

'

(1-39)

This completes the proof. ■

The results of the previous sections suggest that in the case the number of cartel mem­ bers n* varies, there is a conflict between having collusion immune to internal defection, on the one hand, and to the entry of new competitors, on the other. In particular, eqs. (1.12), (1.22) and (1.31) show that, whatever the incumbent’s post-entry reaction is, an increase in

n*

induces, on the one hand, an increase in the minimum length o f the punishment

T *

1.3 Considering entry

32

for which the incentive compatibility constraint holds and a decrease in the critical th resh ­ old for which the participation constraint is satisfied

(Tpci*

^ 2)» on the other. W hether, in the end, this change in the number of firms leads to an increase or to a decrease o f the o p ­ timal length o f the punishment phase

{T\yT2)

depends, as already shown, on the height o f the actual entry sunk costs. In particular, Corollary 1 and 2 have shown that the punish­ ment length increases with the number o f firms if the actual level of the sunk entry cost is sufficiently high. If this is the case, then by Proposition 4 w e also have that this increase in the optimal punishment length induced by an increase in the number of cartel m em bers, will in turn lead to an increase in the percentage of periods the cartel spends in a price-w ar. As will be shown in the empirical part o f the chapter, this seems to have occurred d u rin g the operation o f the JEC cartel.

As a final remark, it should be stressed that as we are interested here in finding o u t the necessary and sufficient conditions for which th e cartel is (internally and externally) stable, entry is, in this framework, treated as a disequilibrium phenomenon. Entry is n e v er expected in equilibrium. W hen it occurs, the enlarged set o f active firms coordinate on a

new

equilibrium taking for granted that entry will not occur any more and an anticom pet­ itive conduct by the cartel members becomes more difficult to attain.28 This is im portant for the interpretation o f the empirical results: since entry is actually observed during th e period under study, this means that the data cannot be fully explained by the equilibrium model. It has to be explained with unexpected changes in exogenous variables such as a d e ­ crease in the sunk costs o f entering the railway business (or to the arrival of a lone potential